THE BAER-SPECKER GROUP
Eoin Coleman1
The Baer-Speker group, P, is the group of funtions from the
natural numbers N into the integers Z. While P is very easy to
dene, it is the soure of a wealth of problems, some of whih have
only reently been solved. The aim of this paper is to present
an introdutory aount of old and new results about P, and to
explore some of the onnetions of this group with other areas of
mathematis, in partiular with the real numbers, innitary logi,
and ombinatorial set theory.
Introdution
The Baer-Speker group P is an innite abelian group under the
addition
(f + g)(n) = f (n) + g(n)
for n 2 N. P ontains the subgroup S, the diret sum of ountably
many opies of Z. In the literature, P is also denoted ZN , or Z! .
Sine all the groups onsidered in this paper are abelian, it will
save spae to adopt the onvention that the term group is short
for abelian group. The textbooks [20℄ and [22℄ are good referenes
for innite abelian group theory. We use the symbols !, !1 , and
2! to denote the ardinal numbers of the natural numbers N, the
rst unountable ardinal, and the ardinal number of the real
numbers R, respetively. For example, P has ardinality 2! , and
S has ardinality !.
1
I am very grateful to the organizers of the Irish Mathematial Soiety Conferene in September 1997 for the opportunity to read a shorter
version of this paper to the partiipants.
9
10
IMS Bulletin 40, 1998
There are four setions in the paper. The rst resumes some
basi material on P; the seond looks at a reently disovered onnetion relating slender subgroups of P to ertain ardinal invariants of the real numbers. In the third setion, some logial aspets
of P are explored. The nal setion is about the omplexity of the
lattie of subgroups of P. Sine the material overed in the paper
is diverse and sattered aross dierent domains, I have not given
many proofs, hoping that the bibliography will enable the reader
to follow themes in more depth.
1. Basis
The struture of innite free groups is relatively lear (see, for
example, [22℄): for eah innite ardinal , there is exatly one
free group of ardinality on generators (up to isomorphism).
So an obvious initial question in the study of P is to determine
how it stands in relation to freeness: is P free?
Denition A group G is free if G is (isomorphi to) a diret sum
of opies of Z.
For example, P has a free subgroup S.
Theorem 1.1 (Baer [1℄) The group P is not free.
We shall dedue this theorem from a stronger assertion below
involving the notion of {freeness.
Denition Suppose that is an innite ardinal. A group G is
{free if every subgroup H G having less than elements is
free.
Every free group is {free for every ardinal , sine subgroups of free groups are free; if < , then {freeness implies {
freeness. Questions about whether {freeness implies {freeness
for < are highly non-trivial and have stimulated one of the
most important researh orientations in innite abelian group theory, leading for example to Shelah's singular ompatness theorem
[7, 25, 20℄ and independene results in set theory [30℄. Sine in
general {freeness is weaker than freeness, we an rene the initial
question about the non-freeness of P and ask whether there are
any ardinals for whih P is {free. Reall that !2 is the seond
unountable ardinal, the ardinal suessor of !1 .
The Baer-Speker Group
11
Proposition 1.2 The group P is not !2 {free.
We need to nd a non-free subgroup of P whih has ardinality !1 . Let p be a xed prime number, and take a pure
subgroup H of ardinality !1 ontaining S and suh that every
element (n1 ; n2 ; : : :) of H has the property that the tail is divisible by arbitrarily high powers of p: (8m)(9r)(8k > r)(pm jnk ).2
Then the quotient group H=pH is a vetor spae over Fp , the nite
eld of p elements. It follows that H is not free, for if H were free,
then H=pH must have dimension (and hene ardinality) !1 ; but
every oset of H=pH ontains an element of S, and hene H=pH
has ardinality at most jSj = ! {a ontradition. So H is not
free.
We an use Proposition 1.2 to improve exerise 19.7 of [22℄:
Proof:
Corollary For every unountable ardinal , there exists a nonfree !1 {free group of ardinality .
Proof: For example, the group < H , the diret sum of opies
of the group H in the proof of Proposition 1.2, will work.
Proposition 1.2 implies Baer's theorem, sine H is a nonfree subgroup of P. However, it leaves open the question whether
ountable subgroups of P are free, i.e. whether P is !1 {free.
Theorem 1.3 (Speker [42℄) The group P is !1 {free.
Proof: This is a well-known result and a full proof is given in
the standard referene textbooks [20, 22℄. It rests on Pontryagin's
Criterion: a ountable group is free if and only if every nite rank
subgroup is free. We shall give a proof of this riterion using logi
in setion three. The proof of Theorem 1.3 proeeds as follows:
every nite rank subgroup of P is embedded in a nitely generated
torsion-free diret summand of P, and hene is free; so if G P is
ountable, then every nite rank subgroup of G is free; now apply
Pontryagin's Criterion.
To lose this setion, let us introdue another type of \loal"
freeness whih has been intensively studied: a group G is almost
free if G is jGj{free (jGj is the number of elements of G), i.e. every
Or: let H be an elementary submodel of the p-adi losure of S in P
of ardinality !1 ontaining S. H exists by the Downward LoewenheimSkolem Theorem of rst-order logi.
2
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IMS Bulletin 40, 1998
subgroup of G of smaller ardinality than G is free. Is P almost
free?
Corollary 1.4 P is almost free if and only if the Continuum
Hypothesis (CH: 2! = !1 ) holds.
If the Continuum Hypothesis is true, then P is almost free if P
is !1 {free, whih is true by Theorem 1.3; if the Continuum Hypothesis is false, then the subgroup H in Proposition 1.2 is a non-free
subgroup of ardinality !1 < 2! = jPj and so P is not almost
free. Thus one annot deide whether P is almost free or not on
the basis of ordinary set theory (ZFC).
One trend in the study of {freeness has been to try to nd
equivalenes between the algebrai and set-theoreti denitions.
In light of Corollary 1.4, it might be interesting to know whether
there are algebrai properties ' and suh that:
(1) P has almost ' i the weak Continuum Hypothesis (wCH:
2! < 2!1 ) holds;
(2) P has almost i Diamond holds.
Diamond is a stronger form of the Continuum Hypothesis CH.
One way to state CH is as a list of guesses A for the subsets of
N:
(9fA : < !1 g suh that
(8X N)(f : X = A g is a stationary subset of !1 )):
A stationary subset of !1 is large: it intersets every losed
unbounded subset of !1 (in the order topology) non-trivially. In
other words, the Continuum Hypothesis says that there is a list of
length !1 whih predits orretly every subset of natural numbers
a large number of times. What about subsets of !1 ? One annot
hope for a list of length !1 whih would predit orretly every
subset of !1 , sine there are 2!1 subsets of !1 and 2!1 > !1 .
But perhaps one might be able to predit orretly just the initial
segments of subsets of !1 . Diamond asserts that there is a list of
!1 guesses A for the initial segments of subsets of !1 and these
guesses are orret on a large subset of !1 . Formally, Diamond
The Baer-Speker Group
13
states:
(9fA : < !1 g suh that
(8X !1 )(f : X \ = A g is a stationary subset of !1)):
A more detailed explanation of this type of predition priniple
an be found in the standard textbooks on set theory [26℄ and
[29℄.
2. P and the real numbers R
We start this setion by looking at some ardinal invariants of the
real numbers R. Reent researh has unovered rather surprising onnetions between these invariants and the size of ertain
subgroups of P.
Denition (1) The additivity of measure, add(L), is the smallest number of measure-zero subsets of R whose union is not of
measure zero.
(2) The additivity of ategory, add(B), is the smallest number
of rst ategory subsets of R whose union is of seond ategory
(not of rst ategory).
(3) The ardinal d, the dominating number, is dened as:
minfjDj : D is a subset of NN and
(8g 2 NN )(9f 2 D)(g(n) f (n) for all but nitely many n)g:
(4) The ardinal b, the bounding number, is dened as:
minfjB j : B is a subset of NN and
(8g 2 NN )(9f 2 B )(g(n) < f (n) for all but nitely many n)g:
The ountable additivity of Lebesgue measure and the Baire ategory theorem imply that add(L) and add(B) are both at least
!1 . And they are also at most 2! . It is immediate too that
!1 b d 2! .
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IMS Bulletin 40, 1998
The reason why the dominating and bounding numbers are
alled invariants of the reals is that the irrationals are homeomorphi to the topologial spae NN when NN is given the produt
topology and N has the disrete topology.
We shall need one other invariant whih is alled the pseudointersetion number. A family F of subsets of N has the strong
nite intersetion property (SFIP) if the intersetion of every
nite subfamily is an innite set. For example, the family of onite subsets of N has the SFIP. A set A is almost ontained
in a set B if A n B is nite.
Denition The ardinal p is
minfjF j : F is a family of subsets of N suh that F has
the SFIP but there is no innite set whih is almost
ontained in every member of F g:
It is an instrutive exerise to show that !1 p 2! .
These ardinals are related as in the following piture, part of
the Cihon diagram (exept for the ardinal p). An arrow relation
! means :
2!
6
d
6
b
6 Miller [32℄
add(L)
6
!1
- add(B)
6 Martin and
Bartoszynski [2℄,
Raisonnier and Stern
[36℄
-
Solovay [31℄
p
The Baer-Speker Group
15
Some of the earliest results on the ardinal invariants of the reals
are due to Rothberger [37℄. The disovery of Martin's Axiom
in 1970 stimulated renewed interest in these and other invariants. A fuller aount of the area whih has been studied in great
depth by mathematiians sine the 1970's is available in the artiles by van Douwen [13℄ and Vaughan [44℄. More reent work has
revealed links between ardinal invariants and quadrati forms:
several of these invariants determine how large orthogonal omplements there are in a quadrati spae. This researh (on Gross and
strongly Gross spaes) is surveyed in the paper [43℄. The bounding number b also appears in reent work (in funtional analysis)
on metrizable barrelled spaes [38℄.
To explain the onnetion of ardinal invariants with the BaerSpeker group P, we shall introdue one further denition.
Denition (Los). A group G is slender if whenever is a homomorphism from P into G, then (en ) = 0 for all but nitely many
n, where en is the element of P that has 1 at the n{th o-ordinate
and 0 everywhere else.
There are several important equivalent haraterizations of
slender groups whih we insert here for the sake of ompleteness
and as an aid to intuition.
Theorem 2.1 (Nunke [33℄, Heinlein [24℄, Eda (1982, see [20℄))
The group G is slender if and only if G does not ontain a opy
of the rationals Q, the yli group of order p Z(p), the p-adi
integers Jp , or P;
equivalently, every homomorphism from P into G is ontinuous,
where G and Z are given the disrete topology and P the produt
topology;
equivalently, for any family fGi : i 2 I g and homomorphism from i2I Gi to G, there are !1 -omplete ultralters D1,: : : , Dn
on I suh that
(8g 2 i2I Gi )(if the support of g; fi 2 I : g(i) 6= 0g;
does not belong to Dk (1 k n); then (g) = 0):
Corollary 2.2 Every !1 {free group whih does not ontain P is
slender.
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IMS Bulletin 40, 1998
Examples The group Z is slender; every free group is slender. P
is not slender (Speker [42℄). Subgroups and diret sums of slender
groups are slender.
Speker's proof that P is not slender works for many other
subgroups of P whih exhibit the Speker phenomenon. But
these subgroups all have ardinality 2! .
Denition (Eda [15℄, Blass [8℄) (1) A subgroup G of P exhibits
the Speker phenomenon i G ontains a sequene fgn : n 2
Ng of linearly independent elements suh that whenever is a
homomorphism from G into Z, then (gn ) = 0 for all but nitely
many n.
(2) The Speker-Eda number, se, is dened as:
minfjGj : G P exhibits the Speker phenomenong:
Corollary 2.3 !1 se 2! .
Theorem 2.4 (Eda [15℄.) (1) CH implies se = !1 .
(2) Martin's Axiom (MA) implies se = 2! .
(3) There is a model of ordinary set theory (ZFC) in whih se <
2! .
In 1994, Andreas Blass observed that Eda's proofs establish
onnetions between the Speker-Eda number and some of the ardinal invariants of the real numbers whih were dened at the
beginning of this setion.
Theorem 2.5 (Eda [15℄, Blass [8℄) (1) p se d.
(2) add(L) se b.
Conjeture 2.6 (Blass [8℄) se = add(B).
3. P and innitary logi
From the logial point of view, a group G is a struture G =
(G; +G ; G ; 0G ) whih satises the axioms for a group. In this
setion we use L to denote the voabulary of groups, that is, L
ontains the onstant, unary, and binary funtion symbols 0, ,
and + to name the zero, inverse, and addition of a group. There
are other possible hoies for the voabulary of groups, but for the
The Baer-Speker Group
17
sake of deniteness we shall x L as above. It is also a fat that
all the theorems of logi presented here are true in muh greater
generality.
The innitary language L1 is the smallest lass of formulas
in the voabulary L whih is losed under negations, onjuntions
of arbitrary length, and strings of quantiers
(9x1 9x2 : : : 9x : : :)(<)
of length less than . This innitary language is more expressive than the rst-order language of groups where one is limited
to negations, nite onjuntions, and nite strings of quantiers.
While the axioms for a group are rst-order, many of the interesting group-theoreti properties annot be expressed by rst-order
sentenes. For example, the following sentene of L1! says that
a group is torsion:
(8g)(g = 0 or 2g = 0 or 3g = 0 or : : : or ng = 0 : : :):
But the onept of torsion annot be axiomatized in a rst-order
language. Innitary logi an express the onepts of {freeness,
{purity, < {generatedness and so on. The paper by Barwise [3℄
is an exellent introdution to the bak and forth methods harateristi of innitary logi. Other useful referenes for innitary logis are the book by Barwise [4℄ and the artile by Dikmann [12℄. A typial problem in general innitary model theory
involves determining whether there are innitarily equivalent nonisomorphi models in various ardinalities, [40℄.
Denition Two groups A and B are L1 {equivalent if and only
if for every sentene ' in L1 : ' is true in A i ' is true in B .
So L1 {equivalent groups annot be distinguished by a sentene in the innitary language L1 . There is an algebrai haraterization of innitary equivalene whih is very useful and perhaps more familiar.
Theorem 3.1 (Karp [27℄, Benda [6℄, Calais [9℄) Two groups A
and B are L1 {equivalent if and only if there is a {extendible
system of partial isomorphisms from A to B .
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IMS Bulletin 40, 1998
A {extendible system of partial isomorphisms from A to B is
a family F of isomorphisms between subgroups of A and B whih
has the {bak-and-forth property: if 2 F is an isomorphism
from M1 A onto N1 B and X (respetively Y ) is a subset
of A (respetively B ) of ardinality less than , then there exists
2 F from M2 onto N2 suh that M1 M2 A, N1 N2 B ,
extends and X (Y ) is a subset of M2 (N2 ).
Using this algebrai onept, it is easy to hek for example
that any two unountable free groups are L1! {equivalent. {
extendible systems are a natural generalization of Cantor's tehnique for showing that there is (up to isomorphism) exatly
one unbounded dense ountable linear order, namely the linearly
ordered set of the rational numbers [3℄. Indeed, for ountable
groups, innitary equivalene and isomorphism are synonymous:
Theorem 3.2 (Sott [39℄) If A and B are ountable L1! {
equivalent groups, then A and B are isomorphi.
The innitary model theory of abelian groups was intensively
studied in the 1970's by Barwise, Eklof, Fisher, Gregory, Kueker,
and Mekler (see [5, 16, 17, 23, 28℄ for example). One of the rst
important results is due to Eklof, who sueeded in determining
whih groups are innitarily equivalent to free groups.
Denition A subgroup A of G is {pure if for every subgroup B
suh that A B G and B=A is < {generated (i.e. generated
by fewer than elements), A is a diret summand of B .
Theorem 3.3 (Eklof [16℄) A group G is L1 {equivalent to a
free group if and only if every < {generated subgroup of G is
ontained in a free, {pure subgroup of G.
For the purposes of this exposition, it is suÆient to know the
following orollaries.
Corollary 3.4 A group G is L1! {equivalent to a free group i
every subgroup of G of nite rank is free.
Eklof used this result to dedue a very famous riterion for
freeness in ountable groups:
Corollary 3.5 (Pontryagin's Criterion [35℄) A ountable group is
free i every subgroup of nite rank is free.
The Baer-Speker Group
Apply Sott's Theorem 3.2 to Corollary 3.4.
Corollary 3.6 (Kueker) A group is L1! {equivalent
19
Proof:
group i it is !1 {free.
to a free
Now we an return to the Baer-Speker group P and see what
these fats tell us.
Corollary 3.7 (Keisler-Kueker) The Baer-Speker group P is
L1! {equivalent to a free group. The lass of free groups is not
denable in L1! .
Corollary 3.8 (Eklof [16℄) The group P is not L1!1 {equivalent
to a free group.
Sine free groups are slender, it follows too from Corollary
3.7 that the lass of slender groups is not denable in L1! . It
might be tempting to onjeture that P is not L1!1 {equivalent
to a slender group. Another possible suggestion is that P is not
L1se {equivalent to a slender group. Mekler showed that if is a
strongly ompat ardinal, then the lass of free groups is denable
in L1 . This prompts the question whether the lass of slender
groups is denable in L1 if is strongly ompat. Eklof and
Mekler have developed appliations of other generalized logis to
problems of abelian group theory in the papers [18℄ and [19℄.
4. The lattie of subgroups of P
The broad thrust in this setion is to desribe some reent researh
on the omplexity of the lattie of subgroups of P. One way to
measure this omplexity is to study what sorts of groups an be
embedded into P. Very generally, the natural questions often have
the form whether there are families of maximal possible size of
subgroups of P whih are strongly dierent (non-isomorphi) in
some preisely dened sense.
An example of this type of theorem in the ontext of general
abelian groups is the following.
Theorem 4.1 (Eklof, Mekler and Shelah [21℄) Under various set-
theoreti hypotheses, there exist families of maximal possible size
of almost free abelian groups whih are pairwise almost disjoint
(the intersetion of any pair ontains no non-free subgroup).
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IMS Bulletin 40, 1998
There is an inverse orrelation between the size of the family
and the strong dierene of its members. If one onsiders families
of pure subgroups of P and takes the notion of strong dierene to
mean that the only homomorphisms between any pair are those of
nite rank, then it is possible to prove the existene of a strongly
dierent family of maximal size.
Theorem 4.2 (Corner and Goldsmith [10℄) Let D be the subgroup
of P ontaining S suh that D=S is the divisible part of P=S.
Let = 2! . There exists a family C onsisting of 2 pure subgroups G of D with D=G rank{1 divisible where eah G is slender,
essentially-indeomposable, essentially-rigid with End(G) = Z +
E0 (G), where E0 (G) is the ideal of all endomorphisms of G whose
images have nite rank.
A similar type of question is the following: does there exist a
family of 2!1 non-isomorphi pure subgroups of P, eah of ardinality !1 , suh that the intersetion of any pair is free? The answer
is positive.
Theorem 4.3 (Shelah and Kolman [41℄) There exists a family
fG : < 2!1 g of pure subgroups of P suh that
(1) eah G has ardinality !1 ;
(2) if 6= , then G \ G is free.
The question whether ertain lasses of group an be embedded in P sometimes leads to independene results. Reall that a
group G is !1 {separable if every ountable subset of G is ontained
in a free diret summand of G.
Theorem 4.4 (Dugas and Irwin [14℄) Embeddability of !1 {
separable groups of ardinality !1 in P is independent of ZFC.
Reexive subgroups of P have also been studied in some
depth. The following result was known for many years under the
additional assumption of the Continuum Hypothesis.
Theorem 4.5 (Ohta [34℄) There exists a non-reexive dual subgroup of P.
The variety displayed in this seletion of results on the BaerSpeker group P illustrates how this easily denable abelian group
is a soure of interesting researh problems whose solutions often
The Baer-Speker Group
21
reveal unexpeted onnetions with problems in other domains of
mathematis.
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IMS Bulletin 40, 1998
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Eoin Coleman,
King's College London,
Strand,
London WC2R 2LS,
England
email: okolmanmember.ams.org